具有有界曲率的黎曼流形上的双调和子流形

On Biharmonic Submanifolds in Riemannian Manifold with Bounded Curvature

利用分部积分法,对截面曲率上界为非负常数的黎曼流形中的完备双调和子流形进行研究。截面曲率上界为非负常数的黎曼流形中的完备双极小子流形,若子流形平均曲率积分满足某种增长性条件时,双调和子流形平均曲率是常数。特别地,单位球面中平均曲率下界为1的完备双调和子流形,若平均曲率积分满足该增长性条件时,则它的平均曲率是1。因而对BMO猜想和S.Meata猜想作出部分肯定的回答。

The complete biharmonic submanifolds in a Riemannian manifolds with sectional curvature bounded from above by a non-negative constant are investigated by using integral by parts. If the integral of their mean curvature satisfies some growth conditions,then their mean curvature is a constant. In particular,the mean curvature of the complete biharmonic submanifolds is bounded as 1 in a sphere and satisfies some growth growth conditions,then the mean curvature is 1. So an affirmative partial answer to BMO conjecture and S. Meata＇s Conjecture is obtained.